For random variables show that
V

Prove that for r .. n

integral(t^r-1 dt

Let X have pdf

x 2

otherwise.

Obtain the cdf of Y X^2.

Let X be a random variable with 13, Use Chebyshev's inequality to obtain P X 8).

Using Central Limit TheJrem, Show that

to 1/2

Let C be a circle of unit area with centre at origin and let S be a square of unit area with
and as the four vertices. If X and y be two independent standard normal variates, show that

J J cKx) cKy) dx dy J cKr) dx dy

where is the pdf of distribution.

Let X follow log-normal with parameters M and cr'2. Find the distribution of Y a 00.

Let be a sequence of pairwise, uncorrelated random variables with
=Mi and cri^2, i
If Oi^2 as then show that

I -fli
0 as
i=1 2
I
cr·
J
j=1

Let X1,X2, ...., be independent Poisson variates with Mi. Find the conditional distribution of X1,..., Xn" |Xi y. 1 to n

Let Y1 denote the first order statistic in a random sample of size n from a distribution that has the pdf e<x<oo
otherwise

Obtain the distribution of Zn

Two points are chosen at random on a tiLe of unit length. Find the probability that each of the 3 line segments will have length greater
than 1/4

Obtain the function of X whose pdf is

A1

-00 x 00

Let be a sequence of random variables with

P(Xn=± 1/2

P(Xn

For what values of a does weak law of large numbers (WLLN) hold?

Three urns U1, U2 and U3 each contain 5 black balls and 7 white balls initially. A ball is drawn at random from U1 and 2 balls of the drawn colour are added to U2. Then a ball is drawn at random from U2 and 3 balls of the drawn colour are added to U3. Find the probability or drawing a white ball from U3.

Let be distributed as bivariate normal BVN 13, 25; Calculate P(4 11.84| X 7).

With 3 variables X1,X2 and X3, it is given that r13 =0.71, R1.23 0·78. Find r12.3.

Suppose the given values of xi are such that a xi b for ..., n. show that 0

Let X have distribution.
Obtain r 0.

If X follows binomial distribution and Y follows provide an appropriate exact test at level a for Ho p1 P2 against H1: P1 P2.

By using Euler -Maclaurin formula, find the sum 1/51^2 1/53^2 ... 1/99^2 .

Given the random samples

Y 10, 12, 18, 20, 26, 28, 32
from the populations hay_ng the distribution function respectively as F1 and F2, test the hypothesis

H0: F1 F2
H0: F1 F2

at level of significance by the Wald -Wolfowitz run test. It is give:! the critical number of runs at sample sizes at level is 5.

Compute Yule's coefficient of association and Yule's coefficient of coligation for the following table
Disease on-set
Yes No
Medicine used A 19 587
B 193 2741

By making use of the difference table and a suitable interpolation formula, find the number of students who obtained less than 45 marks in an examination. from the following table

Marks 30-40 40-50 50-60 60-70 70 -80
Number of Students 31 42 51 35 31

Consider the two samples as follows:
Sample I 10, 12, 14, 16, 23
Sample II 11, 13, 15: 17, 18, 19, 20, 24
Test whether the samples have come from the same populations u31ng Wilcoxon-Mann-Whitney test 10% level of significance. [You can USE normal approximation].

For 20 pairs of heights of father and son(Y) measured in em, the following data were obtained:

x 168.17,E(xi-x)^2 =777.80

(yi 939.42, 9·25 0.932x

Test whether the cut the X-axis err be assumed to be zero, at level of significance

Compute the value of f lnEdx(4 to 5.2) by Simpson's 1/3 rd rule. Given that ln 4.0 1.39, ln 4.2 1.43,
In 4.4 1.48 ln 4.6 1.53, ln 4·8 1.57, ln 5.0 1.61, ln 5.2 1.65

Let X1, X2, ..., Xl2 be a random sample from a normal 02) distribution and Y1,Y2, ..., Y0 be another random sample from normal
independently of each other. carry out an appropriate test for testing

H0 :M1:M2
against

at level of significance It is given that sx^2 y=8 and 15.

Fit the exponential curve y bx to the following data
0 2 4
5.01 10 31·62